Counterparty risk and funding : a tale of two puzzles

380 Pages·2014·6.637 MB·English

by  Stéphane Crépey, Tomasz R Bielecki, Damiano Brigo

Finance
Stéphane Crépey and Tomasz R. Bielecki
Counterparty Risk and Funding
C With an Introductory Dialogue by Damiano Brigo
A Tale of Two Puzzles
o
u
“The landscape of the rates and credit markets has changed so dras- n
tically since the 2008 crisis that older textbooks are barely relevant t
and, from an analytic perspective, appropriate methods have to be e
Counterparty Risk
rethought from scratch. The present volume is one of the best con- r
p
tributions in this direction, featuring a clear description of the various
a
‘value adjustments,’ new models for portfolio credit risk, a unified ana-
r
lytic framework based on BSDEs, and detailed treatment of numerical and Funding
t
methods.” y
—Mark Davis, Professor of Mathematics, Imperial College London
R
“Understanding the subtle interconnections between credit and fund-
i A Tale of Two Puzzles
ing is key to a modern valuation of derivatives. This timely contribu- s
k
tion, written by world-class academics who are also well-recognized
experts in the field, offers a rigorous and comprehensive treatment of
a
the main theories underpinning the new valuation principles. Numerical
n
examples are also provided to help the reader grasp key concepts and
d
ideas of the advanced models and techniques here presented. Overall,

an excellent textbook. Brigo’s dialogue is the icing on the cake.” F
—Fabio Mercurio, Head of Derivatives Research, Bloomberg LP u
n
“A big hooray for this book on CVA, DVA, FVA/LVA, RVA, TVA, and
d
other three letter acronyms.”
—Peter Carr, Global Head of Market Modeling, Morgan Stanley, and i
n
Executive Director, Masters in Math Finance Program, NYU Courant
g
Institute
Features
• Analyzes counterparty risk, funding, and the interaction between
them C
r
é
• Shows how to address the DVA/FVA overlap problem p
w e
• Presents dynamic copula models of portfolio credit risk, including ith y a
the Markov copula common-shock model n
B d
r
• Gives a unified perspective on funding and counterparty risk i B
g
i
o e
models in terms of marked default times l
e
c
k
i

K15071
K15071_Cover.indd 1 4/4/14 9:41 AM
Counterparty Risk
and Funding
A Tale of Two Puzzles
CHAPMAN & HALL/CRC

Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete real-
world examples is highly encouraged.
Series Editors
M.A.H. Dempster Dilip B. Madan Rama Cont
Centre for Financial Research Robert H. Smith School Department of Mathematics
Department of Pure of Business Imperial College
Mathematics and Statistics University of Maryland
University of Cambridge
Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option
Pricing, Pierre Henry-Labordère
Computational Methods in Finance, Ali Hirsa
Counterparty Risk and Funding: A Tale of Two Puzzles, Stéphane Crépey and
Tomasz R. Bielecki, With an Introductory Dialogue by Damiano Brigo
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Mathematics: A Comprehensive Treatment, Giuseppe Campolieti and
Roman N. Makarov
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm,
Ludger Overbeck, and Christoph Wagner
An Introduction to Exotic Option Pricing, Peter Buchen
Introduction to Risk Parity and Budgeting, Thierry Roncalli
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
and Gerald Kroisandt
Monte Carlo Simulation with Applications to Finance, Hui Wang
Nonlinear Option Pricing, Julien Guyon and Pierre Henry-Labordère
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman,
and John J. H. Miller
Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Finance: An Object-Oriented Approach in C++, Erik Schlögl
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug,
and Gautam Mitra
Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Finance: An Introduction with Market Examples, Nicolas Privault
Stochastic Finance: A Numeraire Approach, Jan Vecer
Stochastic Financial Models, Douglas Kennedy
Stochastic Processes with Applications to Finance, Second Edition,
Masaaki Kijima
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm
and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management,
David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
3 Park Square, Milton Park
Abingdon, Oxfordshire OX14 4RN
UK

Counterparty Risk
and Funding
A Tale of Two Puzzles
Stéphane Crépey
University of Evry Val d’Essonne, Evry, France
and
Tomasz R. Bielecki

Illinois Institute of Technology, Chicago, USA
With an Introductory Dialogue by
Damiano Brigo
Imperial College, London, UK
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20140416
International Standard Book Number-13: 978-1-4665-1646-5 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-
ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-
lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-
ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface xv
I FinancialLandscape 1
1 A Galilean Dialogue on Counterparty Risk, CVA, DVA, Multiple Curves,
CollateralandFunding 3
1.1 TotheDiscerningReader . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 TheFirstDay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 General Introduction, Size of Derivatives Markets, Exposures,
CreditVar,Basel . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 TheSecondDay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 CVA, DVA, Pricing, Arbitrage Free Theory, Closeout. And the
Data? Ratings? . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 TheThirdDay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.1 FVA,HardMathswithNoData? CVAVaR,BaselIIIProblems,
CollateralandGapRisk . . . . . . . . . . . . . . . . . . . . . . 30
1.5 TheFourthDay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.1 CounterpartyRiskRestructuring. CCDS,Papillon,FloatingRate
CVAandMarginLending. GlobalCalibration. GlobalValuation.
AvailableCVABooksandForthcomingCVABooks. . . . . . . . 38
2 TheWhysoftheLOIS 47
2.1 FinancialSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 IndifferenceValuationModel . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.1 CreditandFundingCostsSpecification . . . . . . . . . . . . . . 51
2.3 LOISFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 NumericalStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
II Model-FreeDevelopments 61
3 PureCounterpartyRisk 65
3.1 CashFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.1 PromisedDividend . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.2 Collateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.3 CloseoutCashFlow . . . . . . . . . . . . . . . . . . . . . . . . 68
vii
viii Contents
3.2 ValuationandHedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.1 ValuationoftheContract . . . . . . . . . . . . . . . . . . . . . . 70
3.2.2 ValuationofCounterpartyRisk . . . . . . . . . . . . . . . . . . 72
3.2.3 ExposureatDefault . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.4 TVAandCVA/DVA/RC . . . . . . . . . . . . . . . . . . . . . . 74
3.2.4.1 ExpectedPositive/NegativeExposures . . . . . . . . . 75
3.2.4.2 UnilateralCounterpartyRisk . . . . . . . . . . . . . . 75
3.2.5 DynamicHedgingofCounterpartyRisk . . . . . . . . . . . . . . 76
3.2.5.1 Min-VarianceHedging . . . . . . . . . . . . . . . . . 77
3.3 CSASpecifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 CloseOutValuationSchemes . . . . . . . . . . . . . . . . . . . 77
3.3.2 CollateralizationSchemes . . . . . . . . . . . . . . . . . . . . . 78
3.3.3 CurePeriod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.4 RehypothecationRiskandSegregation . . . . . . . . . . . . . . 82
3.3.5 Haircuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.6 CentrallyClearedTrading . . . . . . . . . . . . . . . . . . . . . 83
4 BilateralCounterpartyRiskunderFundingConstraints 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 MarketModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.1 HedgingAssets . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 FundingAssets . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 TradingStrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Self-FinancingCondition . . . . . . . . . . . . . . . . . . . . . . 91
4.3.2 GeneralPrice-and-Hedge . . . . . . . . . . . . . . . . . . . . . . 94
4.4 MartingalePricingApproach . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.1 PrimaryMarket . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2 Q-Price-and-HedgeBSDE . . . . . . . . . . . . . . . . . . . . . 98
4.4.3 Arbitrage,ReplicationandComputationalIssues . . . . . . . . . 100
4.5 TVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 CleanPrice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.2 CSAClose-OutCash-Flow . . . . . . . . . . . . . . . . . . . . . 103
4.5.3 TVARepresentation . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.3.1 CCDSStaticHedgingInterpretation . . . . . . . . . . 105
4.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.2 AnalysisofaSolution . . . . . . . . . . . . . . . . . . . . . . . 107
4.6.2.1 TVA . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.6.2.2 CSAClose-OutPricingSchemes . . . . . . . . . . . . 111
Contents ix
4.6.3 ComparisonwiththeResultsofBurgardandKjaer . . . . . . . . 111
III Reduced-FormBSDEModeling 113
5 AReduced-FormTVABSDEApproachtoCounterpartyRiskunder
FundingConstraints 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Pre-DefaultBSDEModeling . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 BilateralReducedFormSetup . . . . . . . . . . . . . . . . . . . 116
5.2.2 ReductionofFiltration . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.3 ModelingAssumption . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.4 CostProcessesAnalysis . . . . . . . . . . . . . . . . . . . . . . 123
5.3 MarkovCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3.1 FactorProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3.2 Min-VarianceHedgingofMarketRisk . . . . . . . . . . . . . . 126
5.3.3 Min-VarianceHedgingConstrainedtoPerfectHedgingofJump-
to-DefaultRisk . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.4 UnilateralorBilateralintheEnd? . . . . . . . . . . . . . . . . . 133
6 TheFourWingsoftheTVA 135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 TVARepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.2 BSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.2.1 Pre-defaultMarkovSetup . . . . . . . . . . . . . . . . 139
6.2.3 CVA,DVA,LVAandRC . . . . . . . . . . . . . . . . . . . . . . 139
6.3 CSASpecifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3.1 CleanCSARecoveryScheme . . . . . . . . . . . . . . . . . . . 140
6.3.2 Pre-DefaultCSARecoveryScheme . . . . . . . . . . . . . . . . 141
6.3.3 FullCollateralizationCSA . . . . . . . . . . . . . . . . . . . . . 142
6.3.4 PureFunding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.5 AsymmetricalTVAApproach . . . . . . . . . . . . . . . . . . . 142
6.4 CleanValuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4.1 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.2 GaussianVasicekShortRateModel . . . . . . . . . . . . . . . . 146
6.4.2.1 Caplet . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.3 LévyHull-WhiteShortRateModel . . . . . . . . . . . . . . . . 148
6.4.3.1 Caplet . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4.4 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5 TVAComputations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.5.1 TVAEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.5.2 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154